Given data

$P(A)=0.7,\, P(B)=0.4,\, P(A\cap \overline{B})=0.5$

We need to find $P\big(B\mid (A\cup \overline{B})\big).$

Find $P(A\cap B)$

Since $A=(A\cap B)\cup (A\cap \overline{B}),$ and these two parts are disjoint, $P(A)=P(A\cap B)+P(A\cap \overline{B}).$

So, $P(A\cap B)=0.7-0.5=0.2.$

Find the numerator

By definition, $P\big(B\mid (A\cup \overline{B})\big)=\frac{P\big(B\cap (A\cup \overline{B})\big)}{P(A\cup \overline{B})}.$

Now, $B\cap (A\cup \overline{B})=(B\cap A)\cup (B\cap \overline{B}).$

But $B\cap \overline{B}=\varnothing,$ so $B\cap (A\cup \overline{B})=A\cap B.$

Hence numerator: $P\big(B\cap (A\cup \overline{B})\big)=P(A\cap B)=0.2.$

Find the denominator

Use $P(A\cup \overline{B})=P(A)+P(\overline{B})-P(A\cap \overline{B}).$

Since $P(\overline{B})=1-P(B)=1-0.4=0.6,$ we get $P(A\cup \overline{B})=0.7+0.6-0.5=0.8.$

Compute the conditional probability

$P\big(B\mid (A\cup \overline{B})\big)=\frac{0.2}{0.8}=0.25=\frac{1}{4}.$

Match with options

$\frac{1}{4}$ corresponds to Option C.

Comparison with stored answer

Stored correct answer: C

Derived answer: C

So they agree.